Physics Department Centro de Investigación y de Estudios Avanzados del IPN
Low-energy meson phenomenology with Resonance Chiral Lagrangians
Presented in Partial Fulfilment of the Requirements for the Degree of Doctor in Science
by arXiv:1708.00554v1 [hep-ph] 2 Aug 2017 Adolfo Enrique Guevara Escalante
Thesis advisors: Dr. Gabriel López Castro and Dr. Pablo Roig Garcés. ii
“Life is like a healthy penis, it gets hard for no reason.” iii Table of Contents
Table of Contents iii
List of Tables vi
List of Figures viii
1 Theoretical Framework 7 1.1 Introduction ...... 7 1.2 Standard Model ...... 7 1.2.1 Introduction ...... 7 1.2.2 Electroweak Standard Model ...... 8 1.2.3 Strong Interactions and Quantum Chromodynamics ...... 16 1.2.4 Standard Model of Particle Physics ...... 22 1.2.5 QCD, limitations and Effective Field Theories ...... 27 1.2.6 Chiral symmetry of the QCD Lagrangian density ...... 29 1.2.7 Inclusion of external currents ...... 30 1.3 Chiral Perturbation Theory ...... 32 1.3.1 Construction of Chiral Perturbation Theory (χPT) ...... 32 1.4 Resonance Chiral Theory (RχT)...... 35
2 Lepton universality violation and new sources of CP violation 38 2.1 Introduction ...... 38 − − + − 2.2 The τ π ντ ` ` decays as background for BSM interactions . . . 39 → 2.2.1 Introduction ...... 39 2.2.2 Matrix element of the process ...... 40 2.2.3 Form Factors ...... 42 2.2.4 Short distance constraints ...... 45 2.2.5 Branching ratio and invariant mass spectrum ...... 47 2.2.6 Conclusions ...... 51 iv
2.3 Long-distance contribution to B± (π±,K±)`+`− decays...... 52 → 2.3.1 Introduction ...... 52 2.3.2 RχT contribution to the Weak Annihilation amplitude . . . . 53 2.3.3 Extending RχT for heavy flavor mesons ...... 54 2 2.3.4 The electromagnetic form factor FP (q ) ...... 58 2.3.5 CP Asymmetry ...... 65 2.3.6 Conclusions ...... 67
3 New charged current structures 69 3.1 Introduction ...... 69 3.2 Matrix Element and Form Factors ...... 70 3.3 Meson dominance model prediction ...... 72 3.4 Resonance Chiral Theory ...... 78 3.4.1 Resonance Lagrangian operators ...... 78 − − (0) − − (0) 3.5 τ π η γντ as background in the searches for τ π η ντ ... 94 → → 3.5.1 Meson dominance predictions ...... 94 3.5.2 RχL predictions ...... 98 3.6 Statistical error analysis ...... 102 3.7 Conclusions ...... 104
4 The VV 0P form factors in RχT and the π η η0 light-by-light con- − − tribution to the muon g 2 106 − 4.1 Introduction ...... 106 4.2 The anomalous magnetic moment ...... 107 4.3 Hadronic contributions ...... 111 4.4 Transition Form Factor, TFF ...... 113 4.5 η- and η0- Transition Form Factor ...... 116 P,HLbL 4.6 Pseudoscalar exchange contribution aµ ...... 118 4.7 Genuine probe of the πTFF ...... 121 4.8 Conclusions ...... 125 v
5 Conclusions 127
Appendices 130 vi List of Tables
2.1 The central values of the different contributions to the branching ratio of − − + − the τ π ντ ` ` decays (` = e, µ) are displayed on the left-hand side → of the table. The error bands of these branching fractions are given in the right-hand side of the table. The error bar of the IB contribution stems from
the uncertainties on the pion decay constant F and τ` lepton lifetime [72]. . 48 2.2 LD, SD and their interference contributions to the branching ratio for both channels...... 66 2.3 CP asymmetry computed for different q2 ranges, all values are given as percentages...... 66
3.1 Our fitted values of the coupling parameters. Those involving a photon are given multiplied by the unit of electric charge...... 78 3.2 Branching fractions for different kinematical constraints and parameter space points...... 104 3.3 The main conclusions of our analysis are summarized: Our predicted − − (0) branching ratios for the τ π η γντ decays and the corresponding → results when the cut Eγ > 100 MeV is applied. We also compare the latter results to the prediction for the corresponding non-radiative decay (SCC signal) according to ref. [124] and conclude if this cut alone is able to get rid of the corresponding background in SCC searches. . 105
4.1 Different types of contributions to the aµ. The hadronic contributions give the main theoretical uncertainty...... 110
4.2 Contributions to aµ from diagrams (a), (b) and (c) in fig 4.5 as given in ref. [166]...... 113 π0,HLbL 4.3 Our result for aµ in eq. (4.22) is compared to other determinations. The method employed in each of them is also given. We specify those works π0,HLbL that approximate aµ by the pion pole contribution. It is understood that all others consider the complete pion exchange contribution...... 119 vii
P,HLbL 4.4 Our result for aµ in eq. (4.26) is compared to other determinations. The method employed in each of them is also given. We specify those works that P,HLbL approximate aµ by the pseudoscalar pole contribution. It is understood that all others consider the complete pseudoscalar exchange contribution. . 120 HLbL 4.5 Our contribution to the aµ compared to previous computations. . 126 viii List of Figures
+ − 2.1 Feynman diagrams of the different contributions to the τ π` ` ντ → decay. Diagrams (a) to (c) give the model independent contribution, while the structure dependent has been separated into two contribu- tions for convenience ...... 40 2.2 Contribution to the vector form factor in eq (2.1), where the circle with cross denotes the weak vertex...... 43 2.3 Contribution to the axial form factor in eq (2.1), where the circle with cross denotes the weak vertex...... 43 2.4 The different contributions to the normalized e+e− invariant mass distri- bution defined in Eq. (2.15) are plotted. A double logarithmic scale was needed...... 49 2.5 The different contributions to the normalized e+e− invariant mass distribu- 2 tion defined in Eq. (2.15) are plotted in a magnification for s34 & 0.1 GeV intended to better appreciate the SD contributions. A double logarithmic scale was needed...... 50 2.7 All possible contributions to the WA amplitude at leading order in
1/NC . The thick dot denotes interactions between resonances and the fields coupled to the vertex...... 53
2.8 All LD WA Feynman diagrams at leading order in 1/NC . The first row shows the contribution from model independent interactions, while the second and third shows contributions from diagrams with one and two resonances respectively. V (0) stands for light charged (neutral) vector resonances...... 56 2.9 Only non-vanishing structure dependent contribution to the WA LD amplitude...... 58 ix
2.10 BaBar parametrization and our form factor compared with data from p 2 BaBar. Here mll = q . Both form factors overlap below 1.4 GeV, which is the dominant region of the form factor in the observables of the studied decays...... 60 2.11 Electromagnetic form factor of the K meson with the BaBar parametriza-
tion and our form factor compared with data from BaBar. Here mll = p q2...... 61 2 2.12 Real and imaginary parts of FK (q ) using RχT and GS...... 62 2 2.13 Real and imaginary parts of Fπ(q ) using RχT and GS...... 63 2 2.14 The smooth match between LD and QCDf description of FK at 2 GeV is shown...... 64 2 2.15 The smooth match between LD and QCDf description of Fπ at 2 GeV is shown...... 65
3.1 Effective hadronic vertex (grey blob) that defines the Vµν and Aµν tensors. 71 3.2 Photon energy spectra for the leading bremsstrahlung terms in BR(τ → (0) πη γντ ) ...... 72 3.3 Contributions to the effective weak vertex in the MDM model. The wavy line denotes the photon...... 73 3.4 Contributions from the Wess-Zumino-Witten functional [56] to τ − → − π ηγντ decays. The cross circle indicates the insertion of the charged weak current...... 88 3.5 One-resonance exchange contributions from the RχL to the axial- − − vector form factors of the τ π ηγντ decays. Vertices involving → resonances are highlighted with a thick dot...... 89 3.6 Two-resonance exchange contributions from the RχL to the axial- − − vector form factors of the τ π ηγντ decays. Vertices involving → resonances are highlighted with a thick dot...... 89 x
3.7 One-resonance exchange contributions from the RχL to the vector form − − factors of the τ π ηγντ decays. Vertices involving resonances are → highlighted with a thick dot...... 90 3.8 Two-resonance exchange contributions from the RχL to the vector form − − factors of the τ π ηγντ decays. Vertices involving resonances are → highlighted with a thick dot...... 90 − − 3.9 Histogram of BR(τ π ηγντ ) for 100 (left) and 1000 (right) random → points in the MDM parameter space are plotted...... 95 − − 3.10 τ π ηγντ normalized spectra according to MDM in the invariant → mass of the ηπ− system (left) and in the photon energy (right) are plotted for some characteristic points in fig. 3.9 ...... 96
3.11 Histogram of BR(τ πηγντ ) where photons with Eγ > 100 MeV are → rejected...... 96 − − 0 3.12 Histogram of BR(τ π η γντ ) for 100 (left) and 1000 (right) ran- → dom points in the MDM parameter space are plotted...... 97 − − 0 3.13 τ π η γντ normalized spectra according to MDM in the invariant → mass of the π−η0 system (left) and in the photon energy (right) are plotted for some characteristic points in fig. 3.12 ...... 98 0 3.14 Histogram of BR(τ πη γντ ) where photons with Eγ > 100 MeV are → rejected...... 99 − − 3.15 Histogram of BR(τ π ηγντ ) with a sample of 100 RχT parameter → space points for the complete (left) and neglecting 2R diagrams (right) branching fractions...... 99 − − 3.16 τ π ηγντ normalized spectra according to RχT in the invariant → mass of the π−η system (left) and in the photon energy (right) are plotted...... 100
3.17 Histogram of BR(τ πηγντ ) in RχT where photons with Eγ > 100 → MeV are rejected...... 101 xi
− − 0 3.18 Histogram of BR(τ π η γντ ) with a sample of 100 RχT parameter → space points for the complete (left) and neglecting 2R diagrams (right) branching fractions...... 101 − − 0 3.19 τ π η γντ normalized spectra according to RχT in the invariant → mass of the π−η0 system (left) and in the photon energy (right) are plotted...... 102 0 3.20 Histogram of BR(τ πη γντ ) in RχT where photons with Eγ > 100 → MeV are rejected...... 103
4.1 Next to leading order correction to the anomalous magnetic moment found by Schwinger...... 108 4.2 Feynman diagram of a fermion interaction with a classic electromag- netic field. The blob represents all possible interactions that can hap- pen in between...... 109
4.3 Hadronic Vacuum Polarization contribution to aµ, the blob stands for all possible srong interaction processes...... 110 4.4 Light by light scattering insertions for a fermion loop...... 111 4.5 Contributions from Hadronic Light by Light scattering, aµHLbL .... 112 4.6 Main contribution to aHLbL, internal photon lines include the ρ γ µ − mixing ...... 113 4.7 Our fit to the BaBar, Belle, CELLO and CLEO data compared to the Brodsky-Lepage behavior...... 115 4.8 Our prediction for the η (left) and η0 (right) TFF cross section (left) using the couplings of eq. 4.27 compared to BaBar [171], CELLO [173] and CLEO [174]...... 118 4.9 The e+e− µ+µ−π0 scattering as a probe for πTFF with both photons → off-shell...... 121 4.10 Our prediction for σ(e+e− µ+µ−π0) at different center of mass en- → ergies using the couplings in eq. 4.27...... 123 xii
4.11 Our prediction for µ+µ− distribution at s = (1.02 GeV)2 using the couplings in eq. 4.27...... 124 4.12 Our prediction for the σ(e+e− µ+µ−η) cross section (left) and µ+µ− → distribution at 4 GeV2 (right)...... 125 1 Agradecimientos
Agradezco de manera muy sincera a mis asesores, Gabriel López Castro y Pablo Roig Garcés, de quienes he obtenido muchísimos conocimientos y que me han ayu- dado de muchas maneras en mi desarrollo y formación. Gracias a Sally Santiago por ser tan gran soporte para mí en tiempos tan difíciles, en especial el tortuoso lapso de tiempo que pasé sin beca. Agradezco muchísimo a mis sinodales Aurore Courtoy, David Farnández, Iván Heredia, Omar Miranda y Genaro Toledo por su paciencia y ayuda durante mi periodo de formación, en especial en el periodo de revisión de la tesis y el seminario. Agradezco especialmente todas las ayudas recibidas por parte de Eduard de la Cruz. También le doy las gracias al jefe del Departamento, Máximo López (bis) por apoyarme con recursos para que pudiera mostrar mi trabajo en los Rencontres de Moriond.
Debo agradecer también a José Salazar (Chepe), Blanca Cañas (la doctora), Lenin Tostado, Gerardo H. Tomé, Alfonso Jaimes, Jhovanny Mejía, Idalia Sandoval, Bryan Larios y todos los que no logro recordar (y que se quedarán sin ser mencionados por la prisa con la que tuve que escribir esta tesis y no porque no merezcan ser men- cionados); es decir, a todos mis amigos... Y Gus (Gustavo Gutiérrez) por tantas discusiones tan fructíferas en el entendimiento del cómo funciona la naturaleza. Es- peciales agradecimientos a Penguin-San por acompañarnos durante las discusiones de física y otros temas. Gracias, también, a la naturaleza por ser cuántica, debido a que sus fluctuaciones cuánticas a lo largo de la historia del universo me han permitido llegar tan lejos. También, como becario, estoy obligado a dar las gracias a Conacyt por la beca de doctorado, cuyo sistema y personal tienen grandísimas deficiencias que necesitan arreglo urgentemente. Agradezco el apoyo para la obtención de grado por parte del Centro (Cinvestav), así como los apoyos para Curso Especializado con el que asistí a la CERN Latin American School of High Energy Physics y asistencia a Congreso que parcialmente cubrió gastos para asistír a los Rencontres de Moriond. 2 Abstract
It is not known how to obtain exactly transition amplitudes in Quantum Field Theory, so that perturbative approximation is the best we can do. Since the fun- damental theory of strong interactions (Quantum Chromodynamics) does not admit a perturbative approach for processes with energies near or below the proton mass one needs to see how to overcome this difficulty. What common sense dictates is to construct a theory that admits a perturbative description of phenomena at the energy ranges in which the fundamental theory fails to be perturbative. In this thesis we present the computation of some processes that cannot be obtained through an expansion of the strong coupling intensity, since nothing would guarantee the conver- gence of such expansion, this is why we use an Effective Field Theory whose main characteristic is chiral invariance.
On the other hand, since the 1970’s, the Standard Model of fundamental particles interactions has been so successful that it seems very implausible to see phenomena resulting from interactions beyond this theory (with the exception of everything re- lated to neutrino masses) at leading order in perturbation theory. One then relies on precision tests, for which a very good understanding of the interactions is needed. Since many experiments on the High Intensity Frontier begin to take data in the very near future, in order to improve their power of prediction all possible background in the search for Beyond Standard Model effects must be very well understood.
The observables we have computed are contributions within the Standard Model to processes that either need to have a very well described background or that are not very well understood. Two processes are two different τ lepton decays as background − + − − for processes with lepton number and lepton flavor violation such as τ π ` ` ντ → and background for second class currents for the decay τ πηντ . Another process → we computed was the B± P ±`+`−, where P is either a pion or a Kaon. This was → computed in an effort to try to understand the apparent lepton non-universality mea- 3 sured at LHCb, where we obtained a rather large CP asymmetry for the π channel. Finally, we computed the pseudoscalar light-by-light contribution to the anomalous magnetic moment of the muon, giving a more robust analysis of the theoretical un- certainties and compatible with previous results. 4 Resumen
Actualmente no es posible obtener amplitudes de manera exacta usando Teoría Cuántica de Campos, así que lo mejor que se puede hacer es una aproximación pertur- bativa. Ya que la teoría fundamental de las interacciones fuertes (la Cromodinámica Cuántica) no admite una descripción perturbativa para procesos a escalas energéticas cerca o por debajo de la masa del protón se vuelve necesario buscar la forma sortear esta dificultad. Lo que marca la intuición es construir una teoría que permita una descripción perturbativa de fenómenos a escalas de energía en que la teoría funda- mental no puede dar tal descripción. En este sentido, se presenta el cálculo de varios procesos que no pueden ser obtenidos por medio de algunos procesos que no pueden ser obtenidos por medio de una expansión de la intensidad de interacciones fuertes, ya que no se puede garantizar la convergencia de dicha expansión, por lo que hemos recurrido al uso de una Teoría de Campos Efectiva cuya principal característica es la invarianza quiral.
Por otro lado, desde la década de 1970, el Modelo Estándar de partículas fun- damentales ha tenido tanto éxito que parece muy poco probable encontrar algún fenómeno resultante de interacciones más allá de esta teoría (con excepción de todo lo relacionado con las masas de los neutrinos) a primer orden en teoría de perturba- ciones. Entonces se vuelve necesario recurrir a pruebas de precisión, para lo cual se necesita un buen entendimiento de las interacciones. Ya que muchos experimentos en la frontera de la alta intensidad empezarán a tomar datos en un futuro muy cercano, para mejorar su poder predictivo es necesario entender muy bien cualquier posible ruido de fondo en la búsqueda de efectos más allá del Modelo Estándar.
Las observables que calculamos son contribuciones del Modelo Estándar a proceso que, ya sea necesitan tener un ruido de fondo muy bien descrito o no están bien entendidos. Dos de los procesos son dos diferentes decaimientos del leptón τ como ruido de fondo para procesos con violación de número y sabor leptónico como τ − → 5
+ − − π ` ` ντ y el ruido para el descubrimiento de corrientes de segunda clase en el ± decaimiento τ πηντ . Otro proceso que calculamos fue el decaimiento B → → P ±`+`−, donde P = π, K. Esto se calculó como un esfuerzo en tratar de entender la aparente violación de universalidad leptónica medida por LHCb, donde obtuvimos una asimetría de CP grande para el canal del π. Finalmente, calculamos la contribución principal de la dispersión hadrónica luz por luz la momento magnético anómalo del muón, dando un análisis más robusto de la incertidumbre teórica y que es compatible con resultados previos. 6 7 Chapter 1 Theoretical Framework
1.1 Introduction
In this chapter we show the theoretical framework within quantum field theory needed to compute the observables in subsequent chapters. First we give an introduction us- ing a historical approach of the development of the Standard Model of elementary particles. In section 1.3 we develop Chiral Perurbation Theory from the chiral sym- metry of the QCD Lagrangian and its Spontaneous Symmetry Breaking into vectorial SU(3). In section 1.4 we give the main features of Resonance Chiral Theory, lying the foundations to further enlarge the theory to higher chiral orders.
1.2 Standard Model
1.2.1 Introduction
In this section we give a summary of the historical development of the now called Standard Model of elementary particles. In subsection 1.2.2 we follow the develop- ment of the electroweak unification starting with the chiral symmetry of neutrinos up to the Glashow-Weinberg-Salam model of electroweak interactions. In subsection 1.2.3 we show the historical development of strong interactions until Ne’eman and Gell-Mann’s extension of the isospin model, then introduce the concept of partons and the color charge to conclude with the Lagrangian of strong interactions. In sub- section 1.2.4 we briefly summarize CP violation, the Kobayashi-Maskawa scenario and the dates in which the remaining particles of the Standard Model were discov- ered. In subsection 1.2.5 we discuss the limitations of QCD and define the concept of Effective Field Theory. 8
1.2.2 Electroweak Standard Model
Since Ernest Rutherford’s discovery in 1909 that protons were confined in atomic nuclei positively charged [1], the question of how same charge particles can remain together without repelling each other arose. After James Chadwick’s discovery of the neutron in 1932 [2], a strong interaction was hypothesized to explain why the nucleus (a bounded state of protons and neutrons, as suggested by Dmitri Ivanenko [3]) remain bounded, where Werner Heisenberg proposed the isospin model [4]. The next year, Enrico Fermi proposed the existence of a new interaction to explain β-decay [5], later known as weak interaction, where the interacting term came as products of fermion currents µ F ermi = g ψ¯pγ ψn ψ¯eγµψν , (1.1) L where the subindex in each fermion operator ψ denotes the physical field referred to. It also was the first attempt of including the neutrino as a fundamental field. With this and except for gravity, all now known fundamental interactions had been postulated by then at a quantum level.
Fermi’s theory of beta decay only included the proton, neutron (both within the isospin model), electron and neutrino fields as fundamental, but could be very easily extended to include muons (earlier called µ-mesons), heavier baryons and spin zero fields. Also the particles with strangeness (earlier called η-charge) were able to be allocated in a Fermi-like theory.
Since the Fermi theory was not able to predict some nuclear processes involving ∆J = 0 between nuclei, a generalization of Fermi’s theory was sought by considering all linearly independent combinations of Dirac matrices [6], namely 1, γ5, γµ, γµγ5 and i σµν = 2 [γµ, γν], where the squared brackets denotes the commutator. The Lagrangian reads